Download 09. Quantum Mechanics Part I v4 - CREOL

Quantum Mechanics: The Other
Great Revolution of the 20th
Century – Part I
Michael Bass, Professor Emeritus
CREOL, The College of Optics and
Photonics
University of Central Florida
© M. Bass
Revolution


Since Copernicus the word revolution in
a scientific context means a significant
change in how we think.
The 20th century saw two of these:
Relativity
Quantum Mechanics

The time frame for both was
remarkably short – 1900 - 1930
© M. Bass
The players in Part I

Joseph Stephan, Ludwig
Boltzmann, Wilhelm Wein,
Lord Rayleigh, James Jeans,
Max Planck, Johann Jacob
Balmer, J.R. Rydberg, J. J.
Thompson, Ernest
Rutherford, Neils Bohr
© M. Bass
Problems

The nagging facts that classical physics could
not explain:
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the black body spectrum,
the line spectra of atoms,
the fact that atoms were stable,
and others including the specific heat of solids at
low temperatures.
© M. Bass
Spectroscopy primer
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Before going further it is useful to learn a little spectroscopy so that we
share the same vocabulary.
Spectroscopy is the study of the spectral properties of emission or
absorption of various materials (e.g., gases, liquids, solids, plasmas)
In the 19th century spectroscopy was done with prism (and maybe grating)
dispersive elements and photographic emulsions on glass plates.
light source
thin slit
dispersing
prism
lens to focus the
slit onto plane of
emulsion
emulsion
on glass
plate
dark room to
develop the
emulsion
developed plate showing images of
the slit (lines) and maybe some black
body spectra too
© M. Bass
The black body spectrum

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Measure the radiated power of the emission from some hot object
such as a hot filament at various temperatures and, as Joseph
Stephan showed experimentally in 1879 and
Ludwig Boltzmann showed theoretically using thermodynamic
arguments, you will find that it depends on the fourth power of the
filament’s temperature, e.g., P(T) = sT4.
 The Stephan-Boltzmann law: radiated power is proportional
to T4.
In 1894 Wilhelm Wein determined that the wavelength of the
maximum black body radiation times the temperature of the black
body was a constant, e.g. lT=2898 mm K.
Unfortunately, at very long wavelengths (low temperatures)
the experimental data disagreed with this result!!!
© M. Bass
The Rayleigh-Jeans Law
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Rayleigh in 1900 and with Jeans corrections in 1905
formulated the Rayleigh-Jeans Law in which the
energy density of black body radiation per unit
wavelength interval is proportional to the
temperature divided by the fourth power of the
wavelength.
This worked well in the infrared but when re-written in
terms of frequency it suggests that the energy density
increases without bound as the frequency increases.
Well, we knew that black bodies do not radiate
immense amounts of high frequency light and so faced
the ultraviolet catastrophe – failure of the model.
© M. Bass
Enter Max Karl Ernst Ludwig Planck
(1858-1947)


In 1900 Max Planck proposed an empirical explanation of
the black body spectrum. (He did it because it worked.)
Ever since Maxwell’s brilliant theory, everyone accepted
the idea that electromagnetic radiation was a form of wave
phenomenon.
 After all such waves:
 Interfered as waves,
 Diffracted as waves,
 Propagated as waves with speed c
 Were polarized as transverse waves
 Looked, sounded and probably smelled like waves.
© M. Bass
But maybe they weren’t waves

Planck started a revolution by proposing that
in fact they weren’t waves but particles or
quanta of energy with energy proportional to
the frequency that the wave would have.


E = hn
Notice that on the first day of the revolution
we were already confronted with the waveparticle duality that we still suffer with.

We consider waves to have quantum like nature
and particles, quanta, to have wavelike natures
when we need to.
© M. Bass
Assuming light is quantized 

Planck derived his now famous black body radiation spectrum.
By fitting this curve to the data he determined the constants a and b
to be
2
 a = 2pc h where
5 b / lT
h = 6.626 x 10-34 J s, and
 b =hc/ k where
k= Boltzmann’s constant = 1.381 x 10-23 J/K
Note that he did this fitting without the aid of a computer and he
got away with two fitting parameters and one equation.
 He differentiated it to find the peak wavelength and he fit the
derivative to Wein’s law to have a second equation to fit.
He presented this to the Berlin Physical Society on October 19,
1900 – the day on which the revolution began.
Rl  


a
l e
1
1
© M. Bass
If light came in quanta 

Could this account for the line spectra of
atoms?
Line spectra had been a problem for a long
time.
 Look at the sun’s spectrum with a
spectrometer and you find a continuum
broken by missing lines (later recognized as
absorption features of the solar gases).
 Look at the emission of a hydrogen discharge
lamp and you see discrete spectral lines in
the output.
© M. Bass
What was available?
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In the time before the planetary model of atoms, all
that physicists had was the oscillating dipole.
 Remember this was before the electron was
recognized as a particle and before the nucleus
was considered to be made up of protons and
neutrons (as late as 1930 we didn’t even know
about neutrons).
If we get a little ahead of ourselves and consider the
electron-on-a-spring-attached-to-a-stationarynucleus model in the classical sense we can not
account for the lines.
In fact, we can not even account for stable
atoms.
© M. Bass
A classical electron on a spring
atom (no lines and unstable)
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We know from electromagnetics that
accelerating charges radiate energy.
We know that an electron on a spring goes
through accelerated motion and so must
radiate energy.
If it does so it must lose energy and eventually
crash into the nucleus.
Even some sort of specific, characteristic
modes of the oscillating electron would decay
away and so the spectrum before the
crash should be continuous.
In classical physics atoms were not stable and
spectra were continuous. Clearly there was a
problem.
-
+
© M. Bass
The experiments and the data


First you had to prepare atoms of H, N,… not molecules of H2, N2 …
Second you had to perform precision spectroscopy in the days when

photography was barely understood,

emulsions were not sensitive,

emulsion spectral responses were not known,

you did not have CCDs or even Polaroid film,

you had to develop your (home made) plates in the dark
without distorting the emulsion,

then, if you hadn’t yet damaged the emulsion, you had to
measure the distances between the observed spectral lines
with high precision without damaging the plates, and

finally, if you trusted your calibrations, you could report the
wavelengths (frequencies) that you found.
© M. Bass
Balmer
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In 1885, Johann Jacob Balmer, a school teacher in Basel, Switzerland
published a paper in the Proceedings of the Scientific Society of Basel
that was to shake physics in a very fundamental way.
Balmer was not a physicist. He was a numerologist.
 He liked to play with numbers.
He noticed that the wavelengths of the four lines reported by Angstrom
in 1868 could be represented in terms of “a basic number,
h=364.50682 nm, as (9/5)h, (4/3)h,(25/21)h and (9/8)h.”
Then he recognized that these fractions were 9/5, 16/12, 25/21 and
36/32.
2
2
2
2
 The numerators were 3 , 4 , 5 , and 6 , and
2
2
2
2
2
2
2
2
 the denominators were 3 -2 , 4 -2 , 5 -2 and 6 -2 .
2
m
The wavelengths were related by the formula
lh 2 2
m 2
He even predicted other lines in the series and that this
series of lines would end when m = 3 at 656.2 nm.
© M. Bass
Implications
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Somehow the H atom was emitting light with
wavelengths related by integers.
With guidance from a numerologist’s skill with
numbers, physicists went out and determined many
other regularities in spectra and line sequences.
It was clear that integers mattered.
In 1890, J. R. Rydberg claimed he had been using
Balmer’s formula long before it had been published.

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This might have been “sour grapes” but we will never really
know.
Rydberg’s work was much more thorough than Balmer’s and
greatly expanded the utility of the idea that integers mattered
in spectra.
© M. Bass
The formulae worked but
why?
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The situation was similar to classical
mechanics when there was Copernicus’
model, Brahe’s data, Kepler’s formulae,
and Galileo’s concepts.
 The synthesis was missing!!
Then the next big clue came from J. J.
Thompson and others.
 The discovery of the electron
 The wrong model of the atom
Still the electron would spiral in as it
radiated a continuum of light and atoms
would not be stable.
Either something was wrong with the
scattering experiments that pointed to
this model or something was wrong with
classical physics.
© M. Bass
Bohr’s experience

In 1912, after about half a year working in J. J. Thompson’s lab at
Cambridge, Neils Bohr was gently booted out.
 He was booted because he disagreed with Thompson’s model for
the atom.


The plum pudding model
He went to work for Ernest Rutherford (a New Zealander) (the
planetary model) at Manchester.
 This is the first step in the synthesis that would lead to Quantum
Mechanics as we know it.
 Bohr would establish his model for the H atom and it would lead to
the general acceptance of Rutherford’s model of the atom.
© M. Bass
Everything is quantized

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Where Planck introduced the concept of quanta
to electromagnetic energy, Bohr introduced
the concept of quanta to the motion of matter.
 Today we say very loosely that Bohr “quantized
matter”.
 We already had conceived of quanta of charge
and mass so the general idea was not too hard
to accept.
The problem was, “Why should something so
odd be so intrinsically a part of nature?”
© M. Bass
Bohr’s model (a zeroth order picture
of how the atom would work)
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We could picture it but we didn’t know
why it worked as he said.
 Just as in Newton’s day people didn’t
know what forces were but they could
conceive what they did.
Bohr stated the obvious – atoms have E = hn
stable states and when the state of the
atom changes from one of these states
to another a quantum of electromagnetic
energy is released (or absorbed) having
energy equal to the energy difference
between the two states.
This demanded that the states have
energies described by integers.
E = En
E = hn
E = En-1
hn = En – En-1
© M. Bass
Quantization as a tool
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In the process of quantizing the atom
Bohr showed how to avoid the
singularity of the electron spiraling in
and reaching hyper-relativistic speeds
and the infinity of density that the
classical planetary model predicted.
Bohr gave modern theorists the tool of
quantization to remove singularities.
© M. Bass
Previews of Coming
Attractions
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Next time in Quantum Mechanics, Part II we
will:
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Go into the planetary model problem.
Look at Bohr’s model in more depth.
See that Bohr’s derivation of the Rydberg constant
was crucial to the acceptance of Quantum
Mechanics.
Examine the Correspondence Principle.
Discuss quantized angular momentum and spin.
Introduce de Broglie’s waves.
Preview Part III
© M. Bass